**Important Number Systems** have particular basic that are key in the study of computer language for computer science in Kenya.

# Important Number Systems Basics

Number systems as we use them consist of a basic set i.e. the digits, and a base (how many digits). The set of digits in Important Number Systems is denoted by Z and the base by the letter B. E.g. Z = {0,1,2,3,4,5,6,7,8,9}

B = |Z| = 10

A number in Important Number Systems is a linear sequence of digits. The value of a digit at a specific position depends on its assigned meaning and on its position. The value of a number is the sum of these values.

Number: N_{B} = d_{n-1 }d_{n-2}…..d_{1}d_{0} with word length n.

Value: N_{B} = Σd_{1}B^{1} = d_{n-1}B^{n-1}+d_{n-2}B^{n-2}+…..d_{1}B^{1}.

## Useful Number Systems in Important Number Systems for Computers

Name |
Base |
Digits |

Binary | 2 | 0,1 |

Octal | 8 | 0,1,2,3,4,5,6,7 |

Decimal | 10 | 0,1,2,3,4,5,6,7,8,9 |

Hexadecimal sedecimal | 16 | 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F |

Binary data representation in Important Number Systems

The *bit* is the smallest unit of information in Important Number Systems. It has only two options/orders to give i.e. yes/no, on/off, 1/0, 5v/0v etc. The *byte* consists of 8 bits;

2^{8} = 256 different states

Word: this is the number of bits (word length) which can be processed by a computer in a single step (e.g. 32 or 64) →machine dependent.

Representation;

n-1 3 2 1 0

….

2^{n-1} 8 4 2 1

The word size in Important Number Systems for any given computer is fixed. E.g. for a 16-bit word, every word (memory location) can hold a 16-bit pattern with each bit either 0 or 1.

Q: – how many distinct patterns are there in a 16-bit word?

Each bit has two possible values, 0 or 1. Therefore, one bit has two distinct patterns. Its either 0,1 or 1,0. With two bits, each position has two possible values and therefore four distinct patterns i.e. 00,01,10,11. With three bits, each position has two possible values, and thus eight distinct bit patterns i.e. 000,100,001,101,010,110,011,111.

In general, for n bits (a word of length n) in Important Number Systems, we have 2^{n} distinct bit patterns.

Decimal Number systems in Important Number Systems

Remember: Z = {0,1,2,3,4,5,6,7,8,9}, and B = 10;

Each position to the left of a digit in decimal number systems in Important Number Systems increases by a power of ten, and each position to the right of a digit decimal number systems in Important Number Systems decreases by a power of 10. E.g. 47692_{10 }= 2×10^{0 }+ 9×10^{1 }+ 6×10^{2} + 7×10^{3} + 4×10^{4}

Counting in binary

Decimal |
Binary |
Decimal |
Binary |

0 | 00 000 | 16 | 10 000 |

1 | 00 001 | 17 | 10 001 |

2 | 00 010 | 18 | 10 011 |

3 | 00 011 | 19 | 10 100 |

4 | 00 100 | 20 | 10 101 |

5 | 00 101 | 21 | 10 110 |

6 | 00 110 | 22 | 10 111 |

7 | 00 111 | 23 | 11 000 |

8 | 01 000 | 24 | 11 001 |

9 | 01 001 | 25 | 11 011 |

10 | 01 010 | 26 | 11 010 |

11 | 01 011 | 27 | 11 011 |

12 | 01 100 | 28 | 11 100 |

13 | 01 101 | 29 | 11 101 |

14 | 01 110 | 30 | 11 110 |

15 | 01 111 | 31 | 11 111 |

Octal number systems in Important Number Systems

Remember: Z = {0,1,2,3,4,5,6,7} and B = 8;

Each position to the left of a digit in octal number systems in Important Number Systems increases by a power of eight and each position to the right of a digit octal number systems in Important Number Systems decreases by a power of 8. E.g. 12403_{8} = 3×8^{0} + 0×8^{1} + 4×8^{2} + 2×8^{3} + 1×8^{4} = 5379_{10}^{ }

Counting in octal

Decimal |
Octal |
Decimal |
Octal |

0 | 0 0 | 16 | 2 0 |

1 | 0 1 | 17 | 2 1 |

2 | 0 2 | 18 | 2 2 |

3 | 0 3 | 19 | 2 3 |

4 | 0 4 | 20 | 2 4 |

5 | 0 5 | 21 | 2 5 |

6 | 0 6 | 22 | 2 6 |

7 | 0 7 | 23 | 2 7 |

8 | 1 0 | 24 | 3 0 |

9 | 1 1 | 25 | 3 1 |

10 | 1 2 | 26 | 3 2 |

11 | 1 3 | 27 | 3 3 |

12 | 1 4 | 28 | 3 4 |

13 | 1 5 | 29 | 3 5 |

14 | 1 6 | 30 | 3 6 |

15 | 1 7 | 31 | 3 7 |

Hexadecimal number systems in Important Number Systems

Remember: Z = {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} and B = 16

Each position to the left of a digit for hexadecimal number systems in Important Number Systems increases by a power of 16 and each position to the right of a digit in hexadecimal number systems in Important Number Systems decreases by a power of 16.

E.g. FB40A_{16} = 10×16^{0} + 0×16^{1} + 4×16^{2} + 11×16^{3} + 15×16^{4} = 1,029,130_{10}

Counting in hexadecimal

Decimal |
Hexadecimal |
Decimal |
Hexadecimal |

0 | 0 0 | 16 | 1 0 |

1 | 0 1 | 17 | 1 1 |

2 | 0 2 | 18 | 1 2 |

3 | 0 3 | 19 | 1 3 |

4 | 0 4 | 20 | 1 4 |

5 | 0 5 | 21 | 1 5 |

6 | 0 6 | 22 | 1 6 |

7 | 0 7 | 23 | 1 7 |

8 | 0 8 | 24 | 1 8 |

9 | 0 9 | 25 | 1 9 |

10 | 0 A | 26 | 1 A |

11 | 0 B | 27 | 1 B |

12 | 0 C | 28 | 1 C |

13 | 0 D | 29 | 1 D |

14 | 0 E | 30 | 1 E |

15 | 0 F | 31 | 1 F |

### Conclusion of Important Number Systems

The number systems above represent the basics of number systems for computer language. They consist of a basic set (digits) and the base (how many digits) as earlier stated and thus formulate *Important Number Systems* for computers.